Softcover ISBN:  9780821815908 
Product Code:  MMONO/40 
List Price:  $131.00 
MAA Member Price:  $117.90 
AMS Member Price:  $104.80 
Electronic ISBN:  9781470444556 
Product Code:  MMONO/40.E 
List Price:  $131.00 
MAA Member Price:  $117.90 
AMS Member Price:  $104.80 

Book DetailsTranslations of Mathematical MonographsVolume: 40; 1973; 448 ppMSC: Primary 22; Secondary 17;
The contents of this volume are somewhat different from the traditional connotations of the title. First, the author, bearing in mind the needs of the physicist, has tried to make the exposition as elementary as possible. The need for an elementary exposition has influenced the distribution of the material; the book is divided into three largely independent parts, arranged in order of increasing difficulty. Besides compact Lie groups, groups with other topological structure (“similar” to compact groups in some sense) are considered. Prominent among these are reductive complex Lie groups (including semisimple groups), obtained from compact Lie groups by analytic continuation, and also their real forms (reductive real Lie groups). The theory of finitedimensional representation for these classes of groups is developed, striving whenever possible to emphasize the “compact origin” of these representations, i.e. their analytic relationship to representations of compact Lie groups. Also studied are infinitedimensional representations of semisimple complex Lie algebras. Some aspects of the theory of infinitedimensional representations of Lie groups are presented in a brief survey.
Readership 
Table of Contents

Chapters

Preface

Topological groups. Lie groups

Linear groups

Fundamental problems of representation theory

Compact Lie groups. Global theorem

The infinitesimal method in representation theory

Analytic continuation

Irreducible representations of the group $\mathrm {U}(n)$

Tensors and Young diagrams

Casimir operators

Indicator systems and the Gel′fandCetlin basis

Characters

Tensor product of two irreducible representations of $\mathrm {U}(n)$

Basic types of Lie algebras and Lie groups

Classification of compact and reductive Lie algebras

Compact Lie groups in the large

Description of irreducible finitedimensonal representations

Infinitesimal theory (characters, weights, Casimir operators)

Some problems of spectral analysis for finitedimensional representations


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The contents of this volume are somewhat different from the traditional connotations of the title. First, the author, bearing in mind the needs of the physicist, has tried to make the exposition as elementary as possible. The need for an elementary exposition has influenced the distribution of the material; the book is divided into three largely independent parts, arranged in order of increasing difficulty. Besides compact Lie groups, groups with other topological structure (“similar” to compact groups in some sense) are considered. Prominent among these are reductive complex Lie groups (including semisimple groups), obtained from compact Lie groups by analytic continuation, and also their real forms (reductive real Lie groups). The theory of finitedimensional representation for these classes of groups is developed, striving whenever possible to emphasize the “compact origin” of these representations, i.e. their analytic relationship to representations of compact Lie groups. Also studied are infinitedimensional representations of semisimple complex Lie algebras. Some aspects of the theory of infinitedimensional representations of Lie groups are presented in a brief survey.

Chapters

Preface

Topological groups. Lie groups

Linear groups

Fundamental problems of representation theory

Compact Lie groups. Global theorem

The infinitesimal method in representation theory

Analytic continuation

Irreducible representations of the group $\mathrm {U}(n)$

Tensors and Young diagrams

Casimir operators

Indicator systems and the Gel′fandCetlin basis

Characters

Tensor product of two irreducible representations of $\mathrm {U}(n)$

Basic types of Lie algebras and Lie groups

Classification of compact and reductive Lie algebras

Compact Lie groups in the large

Description of irreducible finitedimensonal representations

Infinitesimal theory (characters, weights, Casimir operators)

Some problems of spectral analysis for finitedimensional representations